# Work done on dipole and potential energy in uniform electric field

• ShaunPereira
In summary, the torque on a dipole can be defined asτ=PEsinθThe work done on a dipole to move it from an angle ##\theta_0## to an angle ##\theta_1## can be written as the integral of ##\tau.dr## from ##\theta_0## to ##\theta_1## which is PE[cos##\theta_0##- ##\theta_1## ]The change in potential energy would be ##U_2 -U_1##=##-PEcos\theta_1## -(-##PEcos\theta_0##)=
ShaunPereira
Homework Statement
Finding out relevant equation for work done on a dipole and its potential energy.
Relevant Equations
$$W= -\Delta U$$
$$U= -PEsin\theta$$
I encountered a problem regarding the appropriate sign needed to be taken for the work done on a dipole when it rotates in a uniform electric field and would appreciate some help.

The torque on a dipole can be defined as
τ=PEsinθ
The work done on a dipole to move it from an angle ##\theta_0## to an angle ##\theta_1## can be written as the integral of ##\tau.dr## from ##\theta_0## to ##\theta_1##
which is PE[cos##\theta_0##- ##\theta_1## ]

The change in potential energy would be ##U_2 -U_1##
=##-PEcos\theta_1## -(-##PEcos\theta_0##)
=##PEcos\theta_0##- ##PEcos\theta_1##
=PE[cos##\theta_0##- ##\theta_1## ]

If we compare the equations obtained for work done and the change in potential they both are the same which make sense since the change in potential energy would be the work done.

But this is where my problem begins. Isn't the work done negative of the change in potential energy due to a conservative force?
If yes why is ##W= \Delta U## here and not ##W= -\Delta U##
I know this sound silly but does it make a difference?
I have encountered problems with multiple choices where the sign mattered and wasn't sure which one to choose.

Let's talk about a simpler situation, doing work lifting a brick. If ##F## is the force you apply to the brick, then the work you do on the brick is,

##W = F\cdot \Delta S##

where ##\Delta S## is the vector displacement of the brick. Well, to move the brick up you must push up. Both ##F## and ##\Delta S## are in the same direction so the work is positive. If the brick started at rest and is at rest when you finish lifting it, then ##W## is the change in the potential energy.

If you stop pushing on the brick, it will fall acted upon by a force,

##F = -\nabla U.##

The minus sign makes sense because the force must undo the increase of potential energy put in by lifting.

Paul Colby said:
Let's talk about a simpler situation, doing work lifting a brick. If ##F## is the force you apply to the brick, then the work you do on the brick is,

##W = F\cdot \Delta S##

where ##\Delta S## is the vector displacement of the brick. Well, to move the brick up you must push up. Both ##F## and ##\Delta S## are in the same direction so the work is positive. If the brick started at rest and is at rest when you finish lifting it, then ##W## is the change in the potential energy.

If you stop pushing on the brick, it will fall acted upon by a force,

##F = -\nabla U.##

The minus sign makes sense because the force must undo the increase of potential energy put in by lifting.
Right so if I understood you correctly then what you essentially mean is:
The sign depends on who is doing the work
IF it is by an external agent who lifts the brick, both the the force and the displacement are in the same direction and hence the work done by it is positive and the work done by it is simply the change in potential energy.

However the gravitational force of the Earth on the brick and displacement are in the opposite direction leading to negative work being done by the force which is the negative of change in potential energy.

Please clarify if the above conclusion is right or wrong.

ALSO
and it asked me to find the work done by the external agent to turn the dipole which would essentially be the change in potential energy.

given by $$W=\Delta U$$

Had they asked the work done by the electric field to turn the dipole through some angle I'd assume it would be

$$W=-\Delta U$$

Is this correct?

I think you have it.

ShaunPereira
Ok. Thank You.

## 1. What is a dipole in an electric field?

A dipole is a pair of equal and opposite charges that are separated by a small distance. In an electric field, the dipole experiences a torque, causing it to align with the direction of the field.

## 2. How is work done on a dipole in a uniform electric field?

Work is done on a dipole in a uniform electric field when the dipole is rotated from its initial position to its final position. The work done is equal to the product of the electric field strength, the dipole moment, and the cosine of the angle between them.

## 3. What is the potential energy of a dipole in a uniform electric field?

The potential energy of a dipole in a uniform electric field is the energy required to bring the dipole from infinity to its current position in the field. It is given by the product of the electric field strength, the dipole moment, and the cosine of the angle between them.

## 4. How does the potential energy of a dipole change in a non-uniform electric field?

In a non-uniform electric field, the potential energy of a dipole will change as the dipole moves from one point to another. This change is due to the varying strength of the electric field at different points in the field.

## 5. Can the potential energy of a dipole be negative?

Yes, the potential energy of a dipole can be negative if the angle between the dipole moment and the electric field is greater than 90 degrees. This indicates that the dipole is in a stable equilibrium position in the electric field.

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